Constraint-wrench analysis of robotic manipulators

Abstract

The constraint-wrench analysis of mechanisms, with focus on parallel robots, is the subject of this paper. Although the method proposed here can be generalized for parallel robots with multiple-loop kinematic chains, here, single-loop chains are targeted. To this end, a novel representation of the constraints imposed by the kinematic pairs is introduced. With this representation, the constraint matrix of a mechanism is readily derived. For the calculation of the constraint wrenches, by means of the constraint matrix and based on the Newton–Euler formulation, a new procedure is introduced. As a case study, the constraint wrench analysis of the McGill Schönflies Motion Generator (SMG), while undergoing a test cycle adopted by the industry, is conducted.

Keywords

Notes

Acknowledgements

The authors would like to thank Canada’s Natural Sciences and Engineering Research Council (NSERC) for providing funds to support this research via an Idea to Innovation Grant, which allowed the team to produce the experimental platform motivating the work reported here. Further work has been supported under NSERC’s Discovery Grants program and partly through a James McGill Professorship to the second author.

Appendix A: Derivation of the twist shaping matrices

As shown in Eq. (52), the joint angle vector is mapped onto the twist vector of a robot by means of the twist shaping matrix. This matrix is composed of the twist-shaping matrices Ti of all rigid bodies, namely,

In the sequel, the twist-shaping matrices of the seven rigid bodies of the McGill SMG, along with their time derivatives, will be needed. The parameters regarding the geometry of the McGill SMG that are used in the following formulas are depicted in Figs. 25 and 26.

Appendix B: Derivation of the angular velocity dyads

All the rigid bodies in the McGill SMG move under the same Schönflies displacement subgroup, which contains three translation and one rotation about the z-axis of Fig. 12. Therefore, the angular velocity of each rigid body of the Jth limb is the same, i.e., for the first three rigid bodies, which belong to limb I,

in which Ω is the CPM of the angular velocity vector ω and O is the 3×3 zero matrix. It is important to note that since the Newton–Euler equations for a rigid body are cast in terms of the twist, which is a 6-dimensional array, Ω is defined as a matrix of 6×6. Thus, the seven angular velocity dyads corresponding to the robot rigid bodies are